Perturbative treatment of the luminosity distance

We derive a generalized luminosity distance versus redshift relation for a linearly perturbed Friedmann-Lemaitre-Robertson-Walker (FLRW) metric with two scalar mode excitations. We use two equivalent approaches, based on the Jacobi map and the van Vleck determinant, respectively. We apply the resultant formula to two simple models - an exact FLRW universe and an approximate FLRW universe perturbed by a single scalar mode sinusoidally varying with time. For both models, we derive a cosmographic expansion for dL in terms of z. We comment on the interpretation of our results and their possible application to more realistic cosmological models.